Abstract

Rapid and low-power control over the direction of a radiating light field is a major challenge in photonics and a key enabling technology for emerging sensors and free-space communication links. Current approaches based on bulky motorized components are limited by their high cost and power consumption, while on-chip optical phased arrays face challenges in scaling and programmability. Here, we propose a solid-state approach to beam-steering using optomechanical antennas. We combine recent progress in simultaneous control of optical and mechanical waves with remarkable advances in on-chip optical phased arrays to enable low-power and full two-dimensional beam-steering of monochromatic light. We present a design of a silicon photonic system made of photonic-phononic waveguides that achieves 44° field of view with 880 resolvable spots by sweeping the mechanical wavelength with about a milliwatt of mechanical power. Using mechanical waves as nonreciprocal, active gratings allows us to quickly reconfigure the beam direction, beam shape, and the number of beams. It also enables us to distinguish between light that we send and receive.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
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2017 (2)

Y. Shi, S. Han, and S. Fan, “Optical Circulation and Isolation Based on Indirect Photonic Transitions of Guided Resonance Modes,” ACS Photonics 4, 1639–1645 (2017).
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C. J. Sarabalis, Y. D. Dahmani, R. N. Patel, J. T. Hill, and A. H. Safavi-Naeini, “Release-free silicon-on-insulator cavity optomechanics,” Optica 4, 1147 (2017).
[Crossref]

2016 (5)

C. Wolff, R. Van Laer, M. Steel, B. Eggleton, and C. Poulton, “Brillouin resonance broadening due to structural variations in nanoscale waveguides,” New Journal of Physics 18, 1–9 (2016).
[Crossref]

C. J. Sarabalis, J. T. Hill, and A. H. Safavi-Naeini, “Guided acoustic and optical waves in silicon-on-insulator for Brillouin scattering and optomechanics,” APL Photonics 1, 071301 (2016).
[Crossref]

R. Van Laer, R. Baets, and D. Van Thourhout, “Unifying Brillouin scattering and cavity optomechanics,” Physical Review A 93, 15 (2016).
[Crossref]

M. J. Heck, “Highly integrated optical phased arrays: photonic integrated circuits for optical beam shaping and beam steering,” Nanophotonics 6, 93–107 (2016).
[Crossref]

E. A. Kittlaus, H. Shin, and P. T. Rakich, “Large Brillouin amplification in silicon,” Nature Photonics 10, 463–467 (2016).
[Crossref]

2015 (7)

R. Van Laer, B. Kuyken, D. Van Thourhout, and R. Baets, “Interaction between light and highly confined hypersound in a silicon photonic nanowire,” Nature Photonics 9, 199–203 (2015).
[Crossref]

W. S. Rabinovich, C. I. Moore, R. Mahon, P. G. Goetz, H. R. Burris, M. S. Ferraro, J. L. Murphy, L. M. Thomas, G. C. Gilbreath, M. Vilcheck, and M. R. Suite, “Free-space optical communications research and demonstrations at the US Naval Research Laboratory,” Applied Optics 54, F189 (2015).
[Crossref]

J. C. Hulme, J. K. Doylend, M. J. R. Heck, J. D. Peters, M. L. Davenport, J. T. Bovington, L. A. Coldren, and J. E. Bowers, “Fully integrated hybrid silicon two dimensional beam scanner,” Optics Express 23, 5861 (2015).
[Crossref] [PubMed]

F. Aflatouni, B. Abiri, A. Rekhi, and A. Hajimiri, “Nanophotonic coherent imager,” Optics Express 23, 5117 (2015).
[Crossref] [PubMed]

W. S. Rabinovich, P. G. Goetz, M. Pruessner, R. Mahon, M. S. Ferraro, D. Park, E. Fleet, and M. J. DePrenger, “Free space optical communication link using a silicon photonic optical phased array,” Proc. SPIE 9354, 93540B (2015).

Y. Yang, Y. Ma, H. Guan, Y. Liu, S. Danziger, S. Ocheltree, K. Bergman, T. Baehr-Jones, and M. Hochberg, “Phase coherence length in silicon photonic platform,” Optics Express 23, 16890 (2015).
[Crossref] [PubMed]

R. Van Laer, A. Bazin, B. Kuyken, R. Baets, and D. Van Thourhout, “Net on-chip Brillouin gain based on suspended silicon nanowires,” New Journal of Physics 17, 115005 (2015).
[Crossref]

2014 (3)

D. Melati, A. Melloni, and F. Morichetti, “Real photonic waveguides: guiding light through imperfections,” Advances in Optics and Photonics 6, 156–224 (2014).
[Crossref]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Reviews of Modern Physics 86, 1391–1452 (2014).
[Crossref]

M. Megens, B.-W. Yoo, T. Chan, W. Yang, T. Sun, C. J. Chang-Hasnain, M. C. Wu, and D. A. Horsley, “High-speed 32×32 MEMS optical phased array,” International Society for Optics and Photonics 8977, 89770H (2014).

2013 (6)

D. E. Smalley, Q. Y. J. Smithwick, V. M. Bove, J. Barabas, and S. Jolly, “Anisotropic leaky-mode modulator for holographic video displays,”Nature 498, 313–317 (2013).
[Crossref] [PubMed]

J. Sun, E. Timurdogan, A. Yaacobi, E. S. Hosseini, and M. R. Watts, “Large-scale nanophotonic phased array,” Nature 493, 195–199 (2013).
[Crossref] [PubMed]

B.-W. Yoo, M. Megens, T. Chan, T. Sun, W. Yang, C. J. Chang-Hasnain, D. a. Horsley, and M. C. Wu, “Optical phased array using high contrast gratings for two dimensional beamforming and beamsteering,” Opt. Express 21, 12238–12248 (2013).
[Crossref] [PubMed]

B. Eggleton, C. Poulton, and R. Pant, “Inducing and harnessing stimulated Brillouin scattering in photonic integrated circuits,” Adv. Opt. Photon. 5, 536–587 (2013).
[Crossref]

S. Ghaffari, S. A. Chandorkar, S. Wang, E. J. Ng, C. H. Ahn, V. Hong, Y. Yang, and T. W. Kenny, “Quantum Limit of Quality Factor in Silicon Micro and Nano Mechanical Resonators,” Scientific Reports 3, 3244 (2013).
[Crossref] [PubMed]

S. Boudreau, S. Levasseur, C. Perilla, S. Roy, and J. Genest, “Chemical detection with hyperspectral lidar using dual frequency combs,”Optics express 21, 7411–7418 (2013).
[Crossref] [PubMed]

2012 (4)

O. Fursenko, J. Bauer, A. Knopf, S. Marschmeyer, L. Zimmermann, and G. Winzer, “Characterization of Si nanowaveguide line edge roughness and its effect on light transmission,” Materials Science and Engineering B: Solid-State Materials for Advanced Technology 177, 750–755 (2012).
[Crossref]

J. K. Doylend, M. J. R. Heck, J. T. Bovington, J. D. Peters, M. L. Davenport, L. a. Coldren, and J. E. Bowers, “Hybrid III/V silicon photonic source with integrated 1D free-space beam steering,” Optics Letters 37, 4257 (2012).
[Crossref] [PubMed]

A. Schliesser, N. Picqué, and T. Hänsch, “Mid-infrared frequency combs,” Nat. Photonics 6, 440 (2012).
[Crossref]

P. Rakich, C. Reinke, R. Camacho, P. Davids, and Z. Wang, “Giant Enhancement of Stimulated Brillouin Scattering in the Subwavelength Limit,” Physical Review X 2, 1–15 (2012).
[Crossref]

2011 (4)

J. K. Doylend, M. J. R. Heck, J. T. Bovington, J. D. Peters, L. A. Coldren, and J. E. Bowers, “Two-dimensional free-space beam steering with an optical phased array on silicon-on-insulator,” Optics Express 19, 21595 (2011).
[Crossref] [PubMed]

K. Van Acoleyen, W. Bogaerts, and R. Baets, “Two-dimensional dispersive off-chip beam scanner fabricated on silicon-on-insulator,” IEEE Photonics Technology Letters 23, 1270–1272 (2011).
[Crossref]

K. Van Acoleyen, K. Komorowska, W. Bogaerts, and R. Baets, “One-dimensional off-chip beam steering and shaping using optical phased arrays on silicon-on-insulator,” Journal of Lightwave Technology 29, 3500–3505 (2011).
[Crossref]

H. Elgala, R. Mesleh, and H. Haas, “Indoor optical wireless communication: potential and state-of-the-art,” IEEE Communications Magazine 49, 56–62 (2011).
[Crossref]

2010 (2)

S. Selvaraja, W. Bogaerts, P. Dumon, D. Van Thourhout, and R. Baets, “Subnanometer Linewidth Uniformity in Silicon Nanophotonic Waveguide Devices Using CMOS Fabrication Technology,” IEEE Journal of Selected Topics in Quantum Electronics 16, 316–324 (2010).
[Crossref]

A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Reviews of Modern Physics 82, 1155–1208 (2010).
[Crossref]

2008 (1)

D. Engström, J. Bengtsson, E. Eriksson, and M. Goksör, “Improved beam steering accuracy of a single beam with a 1D phase-only spatial light modulator,”Optics express 16, 18275–18287 (2008).
[Crossref] [PubMed]

2007 (2)

F. V. Laere, S. Member, T. Claes, J. Schrauwen, S. Scheerlinck, W. Bogaerts, and D. Taillaert, “Compact Focusing Grating Couplers for Silicon-on-Insulator Integrated Circuits,” IEEE Photonics Technology Letters 19, 1919–1921 (2007).
[Crossref]

C. K. Wang and W. D. Philpot, “Using airborne bathymetric lidar to detect bottom type variation in shallow waters,” Remote Sensing of Environment 106, 123–135 (2007).
[Crossref]

2005 (1)

S. G. Johnson, M. L. Povinelli, M. Soljačić, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Applied Physics B 81, 283–293 (2005).
[Crossref]

2004 (1)

D. Kedar and S. Arnon, “Urban optical wireless communication networks: the main challenges and possible solutions,” IEEE Communications Magazine 42, S2–S7 (2004).
[Crossref]

2002 (1)

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Physical Review E 65, 066611 (2002).
[Crossref]

2001 (1)

A. Tuantranont, V. M. Bright, J. Zhang, W. Zhang, J. A. Neff, and Y. C. Lee, “Optical beam steering using MEMS-controllable microlens array,” Sensors and Actuators A: Physical 91, 363–372 (2001).
[Crossref]

2000 (2)

K. K. Lee, D. R. Lim, H.-C. Luan, A. Agarwal, J. Foresi, and L. C. Kimerling, “Effect of size and roughness on light transmission in a Si/SiO_2 waveguide: Experiments and model,” Applied Physics Letters 77, 1617 (2000).
[Crossref]

A. Matteo, C. Tsai, and N. Do, “Collinear guided wave to leaky wave acoustooptic interactions in proton-exchanged LiNbO3 waveguides,” IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control 47, 16–28 (2000).
[Crossref]

1996 (1)

B. E. Little, “A variational coupled-mode theory including radiation loss for grating-assisted couplers,” Journal of Lightwave Technology 14, 188–195 (1996).
[Crossref]

1995 (1)

J. M. Elson and J. M. Bennett, “Calculation of the power spectral density from surface profile data,” Appl. Opt. 34, 210 (1995).
[Crossref] [PubMed]

1977 (1)

F. R. Gfeller, “A colinear thin-film acousto-optic scanner,” Journal of Physics D: Applied Physics 10, 1833 (1977).
[Crossref]

1969 (2)

D. Marcuse, “Mode Conversion Caused by Surface Imperfections of a Dielectric Slab Waveguide,” Bell System Technical Journal 48, 3187–3215 (1969).
[Crossref]

D. Marcuse, “Radiation Losses of Dielectric Waveguides in Terms of the Power Spectrum of the Wall Distortion Function,” Bell System Technical Journal 48, 3233–3242 (1969).
[Crossref]

1967 (1)

R. W. Dixon, “Photoelastic Properties of Selected Materials and Their Relevance for Applications to Acoustic Light Modulators and Scanners,” Journal of Applied Physics 38, 5149–5153 (1967).
[Crossref]

1966 (2)

E. Gordon, “A review of acoustooptical deflection and modulation devices,” Proceedings of the IEEE 54, 1391–1401 (1966).
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T. Zhang, J. L. Abellan, A. Joshi, and A. K. Coskun, “Thermal management of manycore systems with silicon-photonic networks,” in “Design, Automation & Test in Europe Conference & Exhibition (DATE), 2014,” (IEEE Conference Publications, New Jersey, 2014), pp. 1–6.

Abiri, B.

F. Aflatouni, B. Abiri, A. Rekhi, and A. Hajimiri, “Nanophotonic coherent imager,” Optics Express 23, 5117 (2015).
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Adler, R.

A. Korpel, R. Adler, P. Desmares, and W. Watson, “A Television Display Using Acoustic Deflection and Modulation of Coherent Light,” Proceedings of the IEEE 54, 1429–1437 (1966).
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F. Aflatouni, B. Abiri, A. Rekhi, and A. Hajimiri, “Nanophotonic coherent imager,” Optics Express 23, 5117 (2015).
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S. Ghaffari, S. A. Chandorkar, S. Wang, E. J. Ng, C. H. Ahn, V. Hong, Y. Yang, and T. W. Kenny, “Quantum Limit of Quality Factor in Silicon Micro and Nano Mechanical Resonators,” Scientific Reports 3, 3244 (2013).
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K. Van Acoleyen, W. Bogaerts, and R. Baets, “Two-dimensional dispersive off-chip beam scanner fabricated on silicon-on-insulator,” IEEE Photonics Technology Letters 23, 1270–1272 (2011).
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S. Selvaraja, W. Bogaerts, P. Dumon, D. Van Thourhout, and R. Baets, “Subnanometer Linewidth Uniformity in Silicon Nanophotonic Waveguide Devices Using CMOS Fabrication Technology,” IEEE Journal of Selected Topics in Quantum Electronics 16, 316–324 (2010).
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O. Fursenko, J. Bauer, A. Knopf, S. Marschmeyer, L. Zimmermann, and G. Winzer, “Characterization of Si nanowaveguide line edge roughness and its effect on light transmission,” Materials Science and Engineering B: Solid-State Materials for Advanced Technology 177, 750–755 (2012).
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R. Van Laer, A. Bazin, B. Kuyken, R. Baets, and D. Van Thourhout, “Net on-chip Brillouin gain based on suspended silicon nanowires,” New Journal of Physics 17, 115005 (2015).
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D. Engström, J. Bengtsson, E. Eriksson, and M. Goksör, “Improved beam steering accuracy of a single beam with a 1D phase-only spatial light modulator,”Optics express 16, 18275–18287 (2008).
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Bennett, J. M.

J. M. Elson and J. M. Bennett, “Calculation of the power spectral density from surface profile data,” Appl. Opt. 34, 210 (1995).
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Y. Yang, Y. Ma, H. Guan, Y. Liu, S. Danziger, S. Ocheltree, K. Bergman, T. Baehr-Jones, and M. Hochberg, “Phase coherence length in silicon photonic platform,” Optics Express 23, 16890 (2015).
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Bogaerts, W.

K. Van Acoleyen, W. Bogaerts, and R. Baets, “Two-dimensional dispersive off-chip beam scanner fabricated on silicon-on-insulator,” IEEE Photonics Technology Letters 23, 1270–1272 (2011).
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K. Van Acoleyen, K. Komorowska, W. Bogaerts, and R. Baets, “One-dimensional off-chip beam steering and shaping using optical phased arrays on silicon-on-insulator,” Journal of Lightwave Technology 29, 3500–3505 (2011).
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S. Selvaraja, W. Bogaerts, P. Dumon, D. Van Thourhout, and R. Baets, “Subnanometer Linewidth Uniformity in Silicon Nanophotonic Waveguide Devices Using CMOS Fabrication Technology,” IEEE Journal of Selected Topics in Quantum Electronics 16, 316–324 (2010).
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F. V. Laere, S. Member, T. Claes, J. Schrauwen, S. Scheerlinck, W. Bogaerts, and D. Taillaert, “Compact Focusing Grating Couplers for Silicon-on-Insulator Integrated Circuits,” IEEE Photonics Technology Letters 19, 1919–1921 (2007).
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Boudreau, S.

S. Boudreau, S. Levasseur, C. Perilla, S. Roy, and J. Genest, “Chemical detection with hyperspectral lidar using dual frequency combs,”Optics express 21, 7411–7418 (2013).
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J. C. Hulme, J. K. Doylend, M. J. R. Heck, J. D. Peters, M. L. Davenport, J. T. Bovington, L. A. Coldren, and J. E. Bowers, “Fully integrated hybrid silicon two dimensional beam scanner,” Optics Express 23, 5861 (2015).
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J. K. Doylend, M. J. R. Heck, J. T. Bovington, J. D. Peters, L. A. Coldren, and J. E. Bowers, “Two-dimensional free-space beam steering with an optical phased array on silicon-on-insulator,” Optics Express 19, 21595 (2011).
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Bowers, J. E.

J. C. Hulme, J. K. Doylend, M. J. R. Heck, J. D. Peters, M. L. Davenport, J. T. Bovington, L. A. Coldren, and J. E. Bowers, “Fully integrated hybrid silicon two dimensional beam scanner,” Optics Express 23, 5861 (2015).
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J. K. Doylend, M. J. R. Heck, J. T. Bovington, J. D. Peters, M. L. Davenport, L. a. Coldren, and J. E. Bowers, “Hybrid III/V silicon photonic source with integrated 1D free-space beam steering,” Optics Letters 37, 4257 (2012).
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J. K. Doylend, M. J. R. Heck, J. T. Bovington, J. D. Peters, L. A. Coldren, and J. E. Bowers, “Two-dimensional free-space beam steering with an optical phased array on silicon-on-insulator,” Optics Express 19, 21595 (2011).
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A. Tuantranont, V. M. Bright, J. Zhang, W. Zhang, J. A. Neff, and Y. C. Lee, “Optical beam steering using MEMS-controllable microlens array,” Sensors and Actuators A: Physical 91, 363–372 (2001).
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W. S. Rabinovich, C. I. Moore, R. Mahon, P. G. Goetz, H. R. Burris, M. S. Ferraro, J. L. Murphy, L. M. Thomas, G. C. Gilbreath, M. Vilcheck, and M. R. Suite, “Free-space optical communications research and demonstrations at the US Naval Research Laboratory,” Applied Optics 54, F189 (2015).
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Byrd, M. J.

C. V. Poulton, A. Yaccobi, Z. Su, M. J. Byrd, and M. R. Watts, “Optical phased array with small spot size, high steering range and grouped cascaded phase shifters,” in “Integrated Photonics Research, Silicon and Nanophotonics,” (Optical Society of America, 2016), pp. IW1B–2.

Camacho, R.

P. Rakich, C. Reinke, R. Camacho, P. Davids, and Z. Wang, “Giant Enhancement of Stimulated Brillouin Scattering in the Subwavelength Limit,” Physical Review X 2, 1–15 (2012).
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Chan, T.

M. Megens, B.-W. Yoo, T. Chan, W. Yang, T. Sun, C. J. Chang-Hasnain, M. C. Wu, and D. A. Horsley, “High-speed 32×32 MEMS optical phased array,” International Society for Optics and Photonics 8977, 89770H (2014).

B.-W. Yoo, M. Megens, T. Chan, T. Sun, W. Yang, C. J. Chang-Hasnain, D. a. Horsley, and M. C. Wu, “Optical phased array using high contrast gratings for two dimensional beamforming and beamsteering,” Opt. Express 21, 12238–12248 (2013).
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S. Ghaffari, S. A. Chandorkar, S. Wang, E. J. Ng, C. H. Ahn, V. Hong, Y. Yang, and T. W. Kenny, “Quantum Limit of Quality Factor in Silicon Micro and Nano Mechanical Resonators,” Scientific Reports 3, 3244 (2013).
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Chang-Hasnain, C. J.

M. Megens, B.-W. Yoo, T. Chan, W. Yang, T. Sun, C. J. Chang-Hasnain, M. C. Wu, and D. A. Horsley, “High-speed 32×32 MEMS optical phased array,” International Society for Optics and Photonics 8977, 89770H (2014).

B.-W. Yoo, M. Megens, T. Chan, T. Sun, W. Yang, C. J. Chang-Hasnain, D. a. Horsley, and M. C. Wu, “Optical phased array using high contrast gratings for two dimensional beamforming and beamsteering,” Opt. Express 21, 12238–12248 (2013).
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F. V. Laere, S. Member, T. Claes, J. Schrauwen, S. Scheerlinck, W. Bogaerts, and D. Taillaert, “Compact Focusing Grating Couplers for Silicon-on-Insulator Integrated Circuits,” IEEE Photonics Technology Letters 19, 1919–1921 (2007).
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Clerk, A. A.

A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Reviews of Modern Physics 82, 1155–1208 (2010).
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Coldren, L. A.

J. C. Hulme, J. K. Doylend, M. J. R. Heck, J. D. Peters, M. L. Davenport, J. T. Bovington, L. A. Coldren, and J. E. Bowers, “Fully integrated hybrid silicon two dimensional beam scanner,” Optics Express 23, 5861 (2015).
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J. K. Doylend, M. J. R. Heck, J. T. Bovington, J. D. Peters, M. L. Davenport, L. a. Coldren, and J. E. Bowers, “Hybrid III/V silicon photonic source with integrated 1D free-space beam steering,” Optics Letters 37, 4257 (2012).
[Crossref] [PubMed]

J. K. Doylend, M. J. R. Heck, J. T. Bovington, J. D. Peters, L. A. Coldren, and J. E. Bowers, “Two-dimensional free-space beam steering with an optical phased array on silicon-on-insulator,” Optics Express 19, 21595 (2011).
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Coskun, A. K.

T. Zhang, J. L. Abellan, A. Joshi, and A. K. Coskun, “Thermal management of manycore systems with silicon-photonic networks,” in “Design, Automation & Test in Europe Conference & Exhibition (DATE), 2014,” (IEEE Conference Publications, New Jersey, 2014), pp. 1–6.

Dahmani, Y. D.

Danziger, S.

Y. Yang, Y. Ma, H. Guan, Y. Liu, S. Danziger, S. Ocheltree, K. Bergman, T. Baehr-Jones, and M. Hochberg, “Phase coherence length in silicon photonic platform,” Optics Express 23, 16890 (2015).
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Davenport, M. L.

J. C. Hulme, J. K. Doylend, M. J. R. Heck, J. D. Peters, M. L. Davenport, J. T. Bovington, L. A. Coldren, and J. E. Bowers, “Fully integrated hybrid silicon two dimensional beam scanner,” Optics Express 23, 5861 (2015).
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J. K. Doylend, M. J. R. Heck, J. T. Bovington, J. D. Peters, M. L. Davenport, L. a. Coldren, and J. E. Bowers, “Hybrid III/V silicon photonic source with integrated 1D free-space beam steering,” Optics Letters 37, 4257 (2012).
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Davids, P.

P. Rakich, C. Reinke, R. Camacho, P. Davids, and Z. Wang, “Giant Enhancement of Stimulated Brillouin Scattering in the Subwavelength Limit,” Physical Review X 2, 1–15 (2012).
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DePrenger, M. J.

W. S. Rabinovich, P. G. Goetz, M. Pruessner, R. Mahon, M. S. Ferraro, D. Park, E. Fleet, and M. J. DePrenger, “Free space optical communication link using a silicon photonic optical phased array,” Proc. SPIE 9354, 93540B (2015).

Desmares, P.

A. Korpel, R. Adler, P. Desmares, and W. Watson, “A Television Display Using Acoustic Deflection and Modulation of Coherent Light,” Proceedings of the IEEE 54, 1429–1437 (1966).
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A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Reviews of Modern Physics 82, 1155–1208 (2010).
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J. C. Hulme, J. K. Doylend, M. J. R. Heck, J. D. Peters, M. L. Davenport, J. T. Bovington, L. A. Coldren, and J. E. Bowers, “Fully integrated hybrid silicon two dimensional beam scanner,” Optics Express 23, 5861 (2015).
[Crossref] [PubMed]

J. K. Doylend, M. J. R. Heck, J. T. Bovington, J. D. Peters, M. L. Davenport, L. a. Coldren, and J. E. Bowers, “Hybrid III/V silicon photonic source with integrated 1D free-space beam steering,” Optics Letters 37, 4257 (2012).
[Crossref] [PubMed]

J. K. Doylend, M. J. R. Heck, J. T. Bovington, J. D. Peters, L. A. Coldren, and J. E. Bowers, “Two-dimensional free-space beam steering with an optical phased array on silicon-on-insulator,” Optics Express 19, 21595 (2011).
[Crossref] [PubMed]

Dumon, P.

S. Selvaraja, W. Bogaerts, P. Dumon, D. Van Thourhout, and R. Baets, “Subnanometer Linewidth Uniformity in Silicon Nanophotonic Waveguide Devices Using CMOS Fabrication Technology,” IEEE Journal of Selected Topics in Quantum Electronics 16, 316–324 (2010).
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C. Wolff, R. Van Laer, M. Steel, B. Eggleton, and C. Poulton, “Brillouin resonance broadening due to structural variations in nanoscale waveguides,” New Journal of Physics 18, 1–9 (2016).
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H. Elgala, R. Mesleh, and H. Haas, “Indoor optical wireless communication: potential and state-of-the-art,” IEEE Communications Magazine 49, 56–62 (2011).
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J. M. Elson and J. M. Bennett, “Calculation of the power spectral density from surface profile data,” Appl. Opt. 34, 210 (1995).
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D. Engström, J. Bengtsson, E. Eriksson, and M. Goksör, “Improved beam steering accuracy of a single beam with a 1D phase-only spatial light modulator,”Optics express 16, 18275–18287 (2008).
[Crossref] [PubMed]

Eriksson, E.

D. Engström, J. Bengtsson, E. Eriksson, and M. Goksör, “Improved beam steering accuracy of a single beam with a 1D phase-only spatial light modulator,”Optics express 16, 18275–18287 (2008).
[Crossref] [PubMed]

Fan, S.

Y. Shi, S. Han, and S. Fan, “Optical Circulation and Isolation Based on Indirect Photonic Transitions of Guided Resonance Modes,” ACS Photonics 4, 1639–1645 (2017).
[Crossref]

Fatemi, R.

R. Fatemi, B. Abiri, and A. Hajimiri, “An 8 × 8 heterodyne lens-less opa camera,” in “CLEO: QELS_Fundamental Science,” (Optical Society of America, 2017), pp. JW2A–9.

Ferraro, M. S.

W. S. Rabinovich, P. G. Goetz, M. Pruessner, R. Mahon, M. S. Ferraro, D. Park, E. Fleet, and M. J. DePrenger, “Free space optical communication link using a silicon photonic optical phased array,” Proc. SPIE 9354, 93540B (2015).

W. S. Rabinovich, C. I. Moore, R. Mahon, P. G. Goetz, H. R. Burris, M. S. Ferraro, J. L. Murphy, L. M. Thomas, G. C. Gilbreath, M. Vilcheck, and M. R. Suite, “Free-space optical communications research and demonstrations at the US Naval Research Laboratory,” Applied Optics 54, F189 (2015).
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Fink, Y.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Physical Review E 65, 066611 (2002).
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Fleet, E.

W. S. Rabinovich, P. G. Goetz, M. Pruessner, R. Mahon, M. S. Ferraro, D. Park, E. Fleet, and M. J. DePrenger, “Free space optical communication link using a silicon photonic optical phased array,” Proc. SPIE 9354, 93540B (2015).

Foresi, J.

K. K. Lee, D. R. Lim, H.-C. Luan, A. Agarwal, J. Foresi, and L. C. Kimerling, “Effect of size and roughness on light transmission in a Si/SiO_2 waveguide: Experiments and model,” Applied Physics Letters 77, 1617 (2000).
[Crossref]

Fursenko, O.

O. Fursenko, J. Bauer, A. Knopf, S. Marschmeyer, L. Zimmermann, and G. Winzer, “Characterization of Si nanowaveguide line edge roughness and its effect on light transmission,” Materials Science and Engineering B: Solid-State Materials for Advanced Technology 177, 750–755 (2012).
[Crossref]

Genest, J.

S. Boudreau, S. Levasseur, C. Perilla, S. Roy, and J. Genest, “Chemical detection with hyperspectral lidar using dual frequency combs,”Optics express 21, 7411–7418 (2013).
[Crossref] [PubMed]

Gfeller, F. R.

F. R. Gfeller, “A colinear thin-film acousto-optic scanner,” Journal of Physics D: Applied Physics 10, 1833 (1977).
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Ghaffari, S.

S. Ghaffari, S. A. Chandorkar, S. Wang, E. J. Ng, C. H. Ahn, V. Hong, Y. Yang, and T. W. Kenny, “Quantum Limit of Quality Factor in Silicon Micro and Nano Mechanical Resonators,” Scientific Reports 3, 3244 (2013).
[Crossref] [PubMed]

Gilbreath, G. C.

W. S. Rabinovich, C. I. Moore, R. Mahon, P. G. Goetz, H. R. Burris, M. S. Ferraro, J. L. Murphy, L. M. Thomas, G. C. Gilbreath, M. Vilcheck, and M. R. Suite, “Free-space optical communications research and demonstrations at the US Naval Research Laboratory,” Applied Optics 54, F189 (2015).
[Crossref]

Girvin, S. M.

A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Reviews of Modern Physics 82, 1155–1208 (2010).
[Crossref]

Goetz, P. G.

W. S. Rabinovich, C. I. Moore, R. Mahon, P. G. Goetz, H. R. Burris, M. S. Ferraro, J. L. Murphy, L. M. Thomas, G. C. Gilbreath, M. Vilcheck, and M. R. Suite, “Free-space optical communications research and demonstrations at the US Naval Research Laboratory,” Applied Optics 54, F189 (2015).
[Crossref]

W. S. Rabinovich, P. G. Goetz, M. Pruessner, R. Mahon, M. S. Ferraro, D. Park, E. Fleet, and M. J. DePrenger, “Free space optical communication link using a silicon photonic optical phased array,” Proc. SPIE 9354, 93540B (2015).

Goksör, M.

D. Engström, J. Bengtsson, E. Eriksson, and M. Goksör, “Improved beam steering accuracy of a single beam with a 1D phase-only spatial light modulator,”Optics express 16, 18275–18287 (2008).
[Crossref] [PubMed]

Gordon, E.

E. Gordon, “A review of acoustooptical deflection and modulation devices,” Proceedings of the IEEE 54, 1391–1401 (1966).
[Crossref]

Guan, H.

Y. Yang, Y. Ma, H. Guan, Y. Liu, S. Danziger, S. Ocheltree, K. Bergman, T. Baehr-Jones, and M. Hochberg, “Phase coherence length in silicon photonic platform,” Optics Express 23, 16890 (2015).
[Crossref] [PubMed]

Haas, H.

H. Elgala, R. Mesleh, and H. Haas, “Indoor optical wireless communication: potential and state-of-the-art,” IEEE Communications Magazine 49, 56–62 (2011).
[Crossref]

Hajimiri, A.

F. Aflatouni, B. Abiri, A. Rekhi, and A. Hajimiri, “Nanophotonic coherent imager,” Optics Express 23, 5117 (2015).
[Crossref] [PubMed]

R. Fatemi, B. Abiri, and A. Hajimiri, “An 8 × 8 heterodyne lens-less opa camera,” in “CLEO: QELS_Fundamental Science,” (Optical Society of America, 2017), pp. JW2A–9.

Han, S.

Y. Shi, S. Han, and S. Fan, “Optical Circulation and Isolation Based on Indirect Photonic Transitions of Guided Resonance Modes,” ACS Photonics 4, 1639–1645 (2017).
[Crossref]

Hänsch, T.

A. Schliesser, N. Picqué, and T. Hänsch, “Mid-infrared frequency combs,” Nat. Photonics 6, 440 (2012).
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Heck, M. J.

M. J. Heck, “Highly integrated optical phased arrays: photonic integrated circuits for optical beam shaping and beam steering,” Nanophotonics 6, 93–107 (2016).
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Heck, M. J. R.

J. C. Hulme, J. K. Doylend, M. J. R. Heck, J. D. Peters, M. L. Davenport, J. T. Bovington, L. A. Coldren, and J. E. Bowers, “Fully integrated hybrid silicon two dimensional beam scanner,” Optics Express 23, 5861 (2015).
[Crossref] [PubMed]

J. K. Doylend, M. J. R. Heck, J. T. Bovington, J. D. Peters, M. L. Davenport, L. a. Coldren, and J. E. Bowers, “Hybrid III/V silicon photonic source with integrated 1D free-space beam steering,” Optics Letters 37, 4257 (2012).
[Crossref] [PubMed]

J. K. Doylend, M. J. R. Heck, J. T. Bovington, J. D. Peters, L. A. Coldren, and J. E. Bowers, “Two-dimensional free-space beam steering with an optical phased array on silicon-on-insulator,” Optics Express 19, 21595 (2011).
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Hill, J. T.

C. J. Sarabalis, Y. D. Dahmani, R. N. Patel, J. T. Hill, and A. H. Safavi-Naeini, “Release-free silicon-on-insulator cavity optomechanics,” Optica 4, 1147 (2017).
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J. Sun, E. Timurdogan, A. Yaacobi, E. S. Hosseini, and M. R. Watts, “Large-scale nanophotonic phased array,” Nature 493, 195–199 (2013).
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M. Megens, B.-W. Yoo, T. Chan, W. Yang, T. Sun, C. J. Chang-Hasnain, M. C. Wu, and D. A. Horsley, “High-speed 32×32 MEMS optical phased array,” International Society for Optics and Photonics 8977, 89770H (2014).

B.-W. Yoo, M. Megens, T. Chan, T. Sun, W. Yang, C. J. Chang-Hasnain, D. a. Horsley, and M. C. Wu, “Optical phased array using high contrast gratings for two dimensional beamforming and beamsteering,” Opt. Express 21, 12238–12248 (2013).
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F. V. Laere, S. Member, T. Claes, J. Schrauwen, S. Scheerlinck, W. Bogaerts, and D. Taillaert, “Compact Focusing Grating Couplers for Silicon-on-Insulator Integrated Circuits,” IEEE Photonics Technology Letters 19, 1919–1921 (2007).
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W. S. Rabinovich, C. I. Moore, R. Mahon, P. G. Goetz, H. R. Burris, M. S. Ferraro, J. L. Murphy, L. M. Thomas, G. C. Gilbreath, M. Vilcheck, and M. R. Suite, “Free-space optical communications research and demonstrations at the US Naval Research Laboratory,” Applied Optics 54, F189 (2015).
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J. Sun, E. Timurdogan, A. Yaacobi, E. S. Hosseini, and M. R. Watts, “Large-scale nanophotonic phased array,” Nature 493, 195–199 (2013).
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A. Matteo, C. Tsai, and N. Do, “Collinear guided wave to leaky wave acoustooptic interactions in proton-exchanged LiNbO3 waveguides,” IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control 47, 16–28 (2000).
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A. Tuantranont, V. M. Bright, J. Zhang, W. Zhang, J. A. Neff, and Y. C. Lee, “Optical beam steering using MEMS-controllable microlens array,” Sensors and Actuators A: Physical 91, 363–372 (2001).
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K. Van Acoleyen, W. Bogaerts, and R. Baets, “Two-dimensional dispersive off-chip beam scanner fabricated on silicon-on-insulator,” IEEE Photonics Technology Letters 23, 1270–1272 (2011).
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K. Van Acoleyen, K. Komorowska, W. Bogaerts, and R. Baets, “One-dimensional off-chip beam steering and shaping using optical phased arrays on silicon-on-insulator,” Journal of Lightwave Technology 29, 3500–3505 (2011).
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R. Van Laer, R. Baets, and D. Van Thourhout, “Unifying Brillouin scattering and cavity optomechanics,” Physical Review A 93, 15 (2016).
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R. Van Laer, R. Baets, and D. Van Thourhout, “Unifying Brillouin scattering and cavity optomechanics,” Physical Review A 93, 15 (2016).
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R. Van Laer, B. Kuyken, D. Van Thourhout, and R. Baets, “Interaction between light and highly confined hypersound in a silicon photonic nanowire,” Nature Photonics 9, 199–203 (2015).
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S. Selvaraja, W. Bogaerts, P. Dumon, D. Van Thourhout, and R. Baets, “Subnanometer Linewidth Uniformity in Silicon Nanophotonic Waveguide Devices Using CMOS Fabrication Technology,” IEEE Journal of Selected Topics in Quantum Electronics 16, 316–324 (2010).
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W. S. Rabinovich, C. I. Moore, R. Mahon, P. G. Goetz, H. R. Burris, M. S. Ferraro, J. L. Murphy, L. M. Thomas, G. C. Gilbreath, M. Vilcheck, and M. R. Suite, “Free-space optical communications research and demonstrations at the US Naval Research Laboratory,” Applied Optics 54, F189 (2015).
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S. Ghaffari, S. A. Chandorkar, S. Wang, E. J. Ng, C. H. Ahn, V. Hong, Y. Yang, and T. W. Kenny, “Quantum Limit of Quality Factor in Silicon Micro and Nano Mechanical Resonators,” Scientific Reports 3, 3244 (2013).
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Yang, W.

M. Megens, B.-W. Yoo, T. Chan, W. Yang, T. Sun, C. J. Chang-Hasnain, M. C. Wu, and D. A. Horsley, “High-speed 32×32 MEMS optical phased array,” International Society for Optics and Photonics 8977, 89770H (2014).

B.-W. Yoo, M. Megens, T. Chan, T. Sun, W. Yang, C. J. Chang-Hasnain, D. a. Horsley, and M. C. Wu, “Optical phased array using high contrast gratings for two dimensional beamforming and beamsteering,” Opt. Express 21, 12238–12248 (2013).
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B.-W. Yoo, M. Megens, T. Chan, T. Sun, W. Yang, C. J. Chang-Hasnain, D. a. Horsley, and M. C. Wu, “Optical phased array using high contrast gratings for two dimensional beamforming and beamsteering,” Opt. Express 21, 12238–12248 (2013).
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A. Tuantranont, V. M. Bright, J. Zhang, W. Zhang, J. A. Neff, and Y. C. Lee, “Optical beam steering using MEMS-controllable microlens array,” Sensors and Actuators A: Physical 91, 363–372 (2001).
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Zhang, W.

A. Tuantranont, V. M. Bright, J. Zhang, W. Zhang, J. A. Neff, and Y. C. Lee, “Optical beam steering using MEMS-controllable microlens array,” Sensors and Actuators A: Physical 91, 363–372 (2001).
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IEEE Photonics Technology Letters (2)

K. Van Acoleyen, W. Bogaerts, and R. Baets, “Two-dimensional dispersive off-chip beam scanner fabricated on silicon-on-insulator,” IEEE Photonics Technology Letters 23, 1270–1272 (2011).
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F. V. Laere, S. Member, T. Claes, J. Schrauwen, S. Scheerlinck, W. Bogaerts, and D. Taillaert, “Compact Focusing Grating Couplers for Silicon-on-Insulator Integrated Circuits,” IEEE Photonics Technology Letters 19, 1919–1921 (2007).
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A. Matteo, C. Tsai, and N. Do, “Collinear guided wave to leaky wave acoustooptic interactions in proton-exchanged LiNbO3 waveguides,” IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control 47, 16–28 (2000).
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M. Megens, B.-W. Yoo, T. Chan, W. Yang, T. Sun, C. J. Chang-Hasnain, M. C. Wu, and D. A. Horsley, “High-speed 32×32 MEMS optical phased array,” International Society for Optics and Photonics 8977, 89770H (2014).

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New Journal of Physics (2)

R. Van Laer, A. Bazin, B. Kuyken, R. Baets, and D. Van Thourhout, “Net on-chip Brillouin gain based on suspended silicon nanowires,” New Journal of Physics 17, 115005 (2015).
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C. Wolff, R. Van Laer, M. Steel, B. Eggleton, and C. Poulton, “Brillouin resonance broadening due to structural variations in nanoscale waveguides,” New Journal of Physics 18, 1–9 (2016).
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T. Zhang, J. L. Abellan, A. Joshi, and A. K. Coskun, “Thermal management of manycore systems with silicon-photonic networks,” in “Design, Automation & Test in Europe Conference & Exhibition (DATE), 2014,” (IEEE Conference Publications, New Jersey, 2014), pp. 1–6.

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C. V. Poulton, A. Yaccobi, Z. Su, M. J. Byrd, and M. R. Watts, “Optical phased array with small spot size, high steering range and grouped cascaded phase shifters,” in “Integrated Photonics Research, Silicon and Nanophotonics,” (Optical Society of America, 2016), pp. IW1B–2.

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Figures (10)

Fig. 1
Fig. 1 (a) A mechanical wave with wavevector K scatters a guided optical wave (β) into free-space (k) at an angle θ. A single antenna scatters light into a cone. Sweeping the mechanical frequency, and therefore K, steers the cone through a range of angles θ. (b) Incorporating these antennas into a phased array forms a beam in the far-field directed into angle ϕ. Antennas consist of partially released silicon ridge waveguides each of which supports guided optical and guided mechanical modes. The displacement field u that scatters guided light (color representing the electric displacement field Dx) out at θ = 60°. (c) Light from a phase-shifter/splitter network (not shown) counterpropagates with mechanical waves driven by a piezoelectric transducer. An interdigital transducer (gray) is patterned on a piezoelectric film (red) deposited on SOI (blue on white). The transducer injects mechanical waves with nanometer-scale displacement and milliwatt-level power into the silicon waveguides.
Fig. 2
Fig. 2 (a) Scattering from the TE-polarized optical mode of a 220 nm thick suspended silicon slab is computed nonperturbatively and the electric displacement field Dy is plotted where the maximal displacement u is set to 30 nm for visibility of the radiated field. The single-sided scattering rate α is quadratic in u even for large displacements. (b) The displacement-normalized scattering rates for the slab and ridge OMA are plotted together for u = 1 nm. Slab rates plotted in black and gray are computed nonperturbatively in 2D (dotted) as well as perturbatively in 2D (dot-dash) and 3D (dashed). For the ridge OMA of section 4, radiation into the optical slab modes (green), air (blue), and the silicon handle (red) add to give the total scattering rate (yellow). (c) Normalizing the scattering rate by power, here 1 mW per antenna for a ridge OMA array and 1 mW per 1.41 µm for the slab, gives a practical measure of the field of view Δθ. Tight confinement of the optical and mechanical energy in the transverse direction for the ridge OMA leads to enhanced radiation for a range of angles compared to the slab case.
Fig. 3
Fig. 3 The ridge OMA guides light and sound supporting a range of mechanical modes over which the 193 THz optical mode (dot on black band of (a) Ey inset) can be scattered from 0° to 90° (shaded in blue). The mechanical power needed to achieve max |uz| = 1 nm oscillations of the core is plotted above the bands of (b). Inset plots of the displacement show the low K behavior (red) where the mode resembles the fundamental Lamb mode of a uniform, clamped membrane. The first excited symmetric mode in the shaded region (black dot) is the mode of interest with scattering rates plotted in Fig. 2. After the anticrossing the mode is expelled from the core into the sockets (blue dot). In (a), the antisymmetric (with respect to y-reflection) and symmetric bands are plotted in black and red, respectively. In (b), symmetric and antisymmetric bands are plotted in black and red, respectively. In both band diagrams, the continuum of radiation modes of the 50 nm SOI stack are hatched.
Fig. 4
Fig. 4 (a) On the transmit side, the optical wave at ω is shifted to an outgoing optical wave at frequency ω + Ω. (b) The backscattered light now has an additional Doppler shift of δω. This incoming wave mixes with the same mechanical wave and excites the optical waveguide at frequency ω + 2Ω + δω. (c) The resulting optical spectrum allows us to distinguish between light being sent and received, moving the received beam outside of the laser phase noise. (d) The outgoing radiation can be multiplexed dynamically by exciting a superposition of mechanical waves. Each of the outgoing beams can be controlled independently. As a special case, this implies widely different optical wavelengths can be sent to and received from the same spot.
Fig. 5
Fig. 5 We compute scattering rates for strong and weak gratings previously demonstrated to verify our numerical methods. The scattering strength scales quadratically with the perturbation u. a, Simulation of a strong SOI grating [47] that has been tested experimentally [47, 48]. The group index of the optical mode is ng = 3.6 and the Bloch index is 2.85. The grating pitch is 630 nm and the core thickness is 220 nm. b, Simulation of a weak grating [49]. The group index of the optical mode is ng = 3.4 and the Bloch index is 3.29. The grating pitch is 10 µm and the core thickness is 500 nm.
Fig. 6
Fig. 6 The scattering rate is insensitive to changes in the PML. a, A sweep of simulation size xend shows initially an increase and then saturation of the scattering rate α when xend > 10 µm. b, The scattering rate drops fast with PML curvature xpml and saturates beyond xpml > 2 µm. c, The scattering rate remains constant when xstart > 3 µm. d, Coarse mesh size does not strongly impact scattering rate below 200 nm. Similarly, the fine mesh size does not affect the scattering rate below 20 nm.
Fig. 7
Fig. 7 The scattering rate αm caused by the antisymmetric sinusoidal perturbation is maximal at thicknesses d = ( l + 1 2 ) λ Si with λ Si = λ n Si = 443 nm the optical wavelength in silicon and l a positive integer, while it vanishes at d = Si. A symmetric sinusoidal perturbation yields the opposite behavior. This behavior is caused by interference between the scattered field from the top surface and that scattered from the bottom surface. As the waveguide core becomes thicker, the structure is less sensitive to either type of perturbation. One simulation point just above a thickness of 600 nm was removed manually as the script had selected the wrong optical mode. We computed both curves for a perturbation u = 10 nm.
Fig. 8
Fig. 8 The scattering rates α are computed for, from top to bottom, a 220 nm, 340 nm, and 440 nm slab with and without the volume current used to implement the JDz component of the boundary interaction. For the 220 nm slab, the J term dominates and our perturbative (dashed) and nonperturbative (dotted) calculations agree well.
Fig. 9
Fig. 9 Mechanical losses alter the exponential envelope of an ideal OMA, limiting the effective aperture and increasing the mechanical power needed to achieve it. a, The solid curves show power radiated per unit length across an antenna for values of ζmax = −log a ranging from −1 to 5 (from red-to-black in even steps). Large scatterings rates a results in approximately exponential radiation patterns as in the lossless case. As a decreases, the profile shifts toward higher z with a constant FWHM of Δζ = 2.45. The dotted curves show the fraction of the guided light remaining in the waveguide, plotted on the same axes. b, The mechanical power needed to achieve a particular radiation pattern is plotted against the antenna’s length L on the same abscissa as a.
Fig. 10
Fig. 10 a, The field of view Δθ is the range of angles that can be reached by tuning the mechanical frequency Ω at fixed optical frequency ω. b, The spot size δθ is the angular width – set by the wavevector uncertainty δK – of the scattered optical beam in the far field. c, The optical bandwidth at each spot Δω is the amount the optical frequency ω can be shifted before the beam angle shifts by more than the spot size δθ at fixed mechanical frequency Ω. Since both the outgoing wavevector k and the guided wavevector β change with frequency, the optical bandwidth at each spot Δω is set by the walk-off between the guided and radiating optical fields.

Tables (4)

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Table 1 Optomechanical antenna properties.

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Table 2 Dephasing in a ridge OMA.

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Table 3 References [47] and [49] provide the scattering rate α as a function u for these gratings. We compute αm by estimating a couple of points in their figures in a similar range of u. Our model agrees with the literature result up to 3% for the strong grating and up to 5.6% for the weak grating.

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Table 4 The scattering rate varies several orders of magnitude depending on the type of perturbation. A symmetric, sinusoidal perturbation has the largest scattering rate in a 220 nm thick silicon slab. For some perturbations the scattering rate scales with u4 instead of u2 because of destructive cancellations in the term proportional u2. In general the scattering rate contains all terms u2l with l any positive integer. Here we use the perturbation-normalized quality factor as Qm = ω/(αmvg).

Equations (60)

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ω r = ω + Ω
k cos θ = β K ,
( × × μ ε ω 2 ) E r = i ω μ J om
J om = i ω ( δ u ε u ) E
P ( z ) = P 0 e α m u 2 z .
J om = i ω Δ ε E y u x y ^ ,
δ φ 2 ( z ) = S β β [ 0 ] z + S K K [ 0 ] ( L z )
δ X l ( Δ z ) δ X l ( 0 ) = σ l 2 e | Δ z | / ξ l
L φ = 2 π 2 S β β [ 0 ] + S K K [ 0 ]
α = κ v g = ω Q v g
t = z P P s
Q = ω κ and κ = P s ,
z P = ω Q
z P = ω Q v g α P .
J n = ( | x | x start x pml ) 2
d = ( l + 1 2 ) λ Si
d = l λ Si
α tot α mb = ( 1 α pe α mb ) 2 = 1 0.26 + 0.29
n core = ( n Si + Δ n x x 0 Δ n x z 0 n Si + Δ n y y 0 Δ n x z 0 n Si + Δ n z z )
Δ n x x = 1 2 n Si 3 p 12 S z z
Δ n y y = 1 2 n Si 3 p 12 S z z
Δ n z z = 1 2 n Si 3 p 11 S z z
Δ n x z = n Si 3 p 44 S x z
S z z = z u z = K 2 x u sin K z
S x z = 1 2 ( z u x + x u z ) = S z x = 0
ε ε + δ u ε u + O ( u 2 ) .
( × × μ ε ω 2 ) E = i ω μ J
( × × μ ε ω 2 ) Θ E = μ ω 2 ( δ u ε u ) E .
Θ E r = μ ε ω 1 2 E + μ ω 2 ( δ u ε u ) E
ω 1 = ω 2 E | δ u ε u | E E | ε | E
Θ E r = μ ω 2 ( δ u ε u ) E .
J om = i ω ( δ u ε u ) E
Θ E r = i ω μ J om .
δ u ε rp u = ( u n ) δ Ω ( Δ ε Π ε Δ ε 1 ε Π )
( δ u ε rp u ) E = u z δ ( z ) [ Δ ε E x Δ ε E y ε Δ ε 1 D z ] .
δ u ε pe u = ε pS ε ε .
n × Δ H = J .
δ φ ( z ) = 0 z d z δ β ( z ) L z d z δ K ( z )
δ φ 2 ( z ) = 0 z 0 z d z d z δ β ( z ) δ β ( z ) + L z L z d z d z δ K ( z ) δ K ( z ) + 2 0 z 0 L d z d z δ β ( z ) δ K ( z ) .
δ X l ( Δ z ) δ X l ( 0 ) = σ l 2 e | Δ z | / ξ l .
0 z 0 z d z d z e | Δ z | / ξ l = 2 ξ l ( z ξ l + ξ l e z / ξ l ) z ξ l 2 ξ l z
δ φ 2 ( z ) = l 2 l β 2 σ l 2 ξ l z + 2 l K 2 σ l 2 ξ l ( L z ) .
S β β [ ω ] = + d z e i ω z δ β ( z ) δ β ( 0 ) .
δ φ 2 ( z ) = S β β [ 0 ] z + S K K [ 0 ] ( L z )
S β β [ ω ] = l 2 ( l β σ l ) 2 ξ l 1 + ( ω ξ l ) 2 .
δ φ ( z ) = 0 z d z δ k
δ φ 2 ( z ) = S k k [ 0 ] z
z P = α m P m P
z P m = γ P m
P ( z ) P 0 = e a ( e ζ 1 )
P r ( ζ ) γ P 0 = a exp ( ζ a ( e ζ 1 ) ) .
k ( ω r + Δ Ω ) cos ( θ + Δ θ ) k ( ω r ) cos ( θ ) = Δ K
Δ θ = Δ Ω k sin ( θ ) ( 1 v m + cos ( θ ) c )
Δ Ω k sin ( θ ) v m = Δ Ω 2 π λ sin ( θ ) v m
k δ cos ( θ ) k sin ( θ ) δ θ = δ K = 2 π L eff
δ θ = λ sin ( θ ) L eff
N θ = Δ K δ k = Δ θ δ θ = Δ Ω 2 π τ m .
cos θ δ k + k δ cos θ = δ K = δ β .
Δ ω 2 π = 1 τ τ r
Δ Ω m 2 π = 1 τ m